A269833 - OEIS

%I #27 Jun 27 2022 16:20:48

%S 0,4,6,8,16,20,21,40,45,47,52,64,67,71,72,74,88

%N Numbers n such that 2^n + n! is the sum of 2 squares.

%C Integers n such that the equation 2^n + n! = x^2 + y^2 where x and y are integers is solvable.

%C 4, 8, 16 and 64 are powers of 2. What is the next power of 2 (if any) in this sequence?

%C 103 <= a(18) <= 108. 108, 117, 144, 176, 254, 537 are terms. - _Chai Wah Wu_, Jul 22 2020

%e 6 is a term because 2^6 + 6! = 28^2.

%e 8 is a term because 2^8 + 8! = 24^2 + 200^2.

%e 21 is a term because 2^21 + 21! = 1222129664^2 + 7042537984^2.

%t Select[Range[0, 64], SquaresR[2, 2^# + #!] > 0 &] (* _Michael De Vlieger_, Mar 07 2016 *)

%o (PARI) isA001481(n) = #bnfisintnorm(bnfinit(z^2+1), n);

%o for(n=0, 1e2, if(isA001481(n!+2^n), print1(n, ", ")));

%o (Python)

%o from math import factorial

%o from itertools import count, islice

%o from sympy import factorint

%o def A269833_gen(): # generator of terms

%o     return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint((1<<n)+factorial(n)).items()),count(0))

%o A269833_list = list(islice(A269833_gen(),9)) # _Chai Wah Wu_, Jun 27 2022

%Y Cf. A001481, A007611.

%K nonn,more

%O 1,2

%A _Altug Alkan_, Mar 06 2016

%E a(17) from _Chai Wah Wu_, Jul 22 2020